Metaphysics & mathematics

Now I have claimed – as a great many other thinkers, both secular and religious, would claim – that philosophy, and in particular the branch of philosophy called metaphysics, is another form of inquiry which is both rational and at least in part non-empirical. It can be thought of as being similar to both empirical science and mathematics in some respects, and different from both in other respects. Like empirical science, metaphysics often begins with things we know via observation. But like mathematics, it arrives at conclusions which, if the reasoning leading to them is correct, are necessary truths rather than contingent ones, truths that could not have been otherwise. That doesn’t mean that the metaphysician is infallible, any more than the mathematician is. It means instead that if he has done his job well, he will (like the mathematician) have discovered truths about the world that are even deeper and more indubitable than the most solid findings of empirical science.

1) The last sentence takes sides in a debate within mathematical philosophy as to the nature of mathematics. A minor point, but I think not trivial.

2) I don’t grant that metaphysics is very analogous to mathematics at all.* There is a reason that powerfully predictive sciences such as physics use mathematics in preference to verbal reasoning. Humans are really bad at reasoning without the formal structure of mathematics. Really bad. Mathematics straitjackets human cleverness, and prevents one from slowly inching toward their preferred conclusion through a sequence of plausible, if not definite, chain of propositions.

Natural science & mathematics know progress. We cede to them pride of place in intellectual disciplines precisely because we see their fruit all around us. The method of mathematical proof is so robust that Euclid’s The Elements is still used as a textbook today because it is of more than historical interest. And yet mathematics does move on at the same time, the elementary techniques learned by most students in the natural sciences (e.g., introductory calculus) are no longer of great intellectual interest.

Note: I do on occasion enjoy reading pre & early modern metaphysicians, but only because of their relevance to the history of thought.

* On second thought, perhaps it is analogous. When I was last interested in philosophical apologetics I would run into a fair amount of logical notation. But, the relation is similar to that of particle physics and social physics (i.e., a quantitative understanding of the dynamics of human societies). I hope that one day social physics can add some genuine value, but it is not even a shadow of what particle physics is.

11 comments

Interesting note by Feser. Of course, since modern philosophy (especially metaphysics) is little more than pseudo-intellectual (or even, sometimes, actually intellectual) games playing, the attempt to glom onto the legitimacy of mathematics is all too predictable. The fact that mathematics is non-empirical, but rational does not mean that all non-empirical thinking, like religious and philosophic thinking, is also rational. They’re not, mathematics is.

I think Feser’s right that philosophy can be mathematical, but we have to understand that mathematics is just a method. One can construct logical proofs of philosophical claims (and this is how modern philosophy is done, by and large), but math and logic in themselves tell you nothing about which axioms you ought to be applying their methods to. At root, most of what we call ‘mathematics’ has solidly empirical roots – Euclidean geometry was all we had for a very long time because its axioms seemed to be the ‘axioms’ of the observed world.

Two things prevent philosophy from being like mathematics. First, the axioms of philosophy aren’t as obviously relevant to the real world. No one thinks that, in everyday life, two points might not lie on a line. But lots of people disagree with similarly basic philosophical claims. One might be able to reason certainly from such claims, but your argument is only sound if the assumptions are good. Second, math has its own language. To get anything useful out of a definitely valid philosophical argument, you have to translate something written in a mathematical language into English, and there’s plenty of opportunity for mistakes in that.

I meant things like the principle of sufficient reason, for example. Or assumptions about morality – it’s not like utilitarians or divine command theorists claim to have rock-solid logical arguments as to why the good is what they say it is. A lot of philosophical argument, in my experience, is about identifying an intuition that you share with someone else and trying to show them that something else follows from that intuition. I think there’s pretty wide agreement that there are multiple consistent but mutually exclusive belief systems, and we don’t have any real way to choose among them other than going with what most people seem to think (consider the problem of induction).

Mathematics is not non-empirical. The confusion arises because many people do not think carefully about what mathematicians actually study, which is the interrelationships of concepts.

The only way concepts can be studied is by examining a physical system that embodies them. The physical system most commonly used by mathematicians is the human brain, and the method of observation is to give parts of their brains input and observe their output with other parts.

The only way to perform mathematics is to process mathematical statements and see what the result is, then form predictions about future results. If we complete the process and get the same result every time, there’s a good bet we’ll get it the next time we evaluate – and that others will get the same result, too.

This process is not free from error. It is not invulnerable to lacking critical data. It does not result in certain, absolute knowledge, and can in fact be overthrown by later evidence. It is a subset of the scientific process.

Well, I went to amazon.com, resigned to order Feser’s latest book – so I could take him on. But the strangest thing happened. I just couldn’t make myself click the “2-day 1-click” button, to instantly purchase it. I read the entire Editorial Reviews section ( quite snarky ), and if they give anything close to a summary of what the book is about, I would just feel a complete fool giving $17.82 of my own money to Amazon, for what I know is going to be several hours of my life I’ll never be able to get back.

“Natural science & mathematics know progress. We cede to them pride of place in intellectual disciplines precisely because we see their fruit all around us.”

Whatever it is the poster is trying to express, he undoes himself with this statement. If the merits of mathematics are expressed by its ability to “progress” and bear “fruit,” then it is as subject to bias as anything else.